lecture-ch4 Motion in Two and Three Dimensions

lecture-ch4 Motion in Two and Three Dimensions - Contents 4...

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Unformatted text preview: Contents 4 Motion in Two and Three Dimensions 1 4.1 Position and Displacement . . . . . . . . . . . . . . . . . . . . 1 4.2 Average Velocity and Instantaneous Velocity . . . . . . . . . . 2 4.3 Average Acceleration and Instantaneous Acceleration . . . . . 2 4.4 Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . . 3 4.4.1 The equation of the Path . . . . . . . . . . . . . . . . . 4 4.4.2 The Horizontal Range . . . . . . . . . . . . . . . . . . 4 4.5 Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.5.1 Uniform Circular Motion . . . . . . . . . . . . . . . . . 6 4.5.2 Another Derivation of Centripetal Acceleration . . . . 7 4.6 Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Motion in Two and Three Dimensions 4.1 Position and Displacement One general way of locating a particle is with a position vector vector r , which is a vector that extends from a reference point (usually the origin of a coordinate system) to the particle. vector r may be written as vector r = x ˆ i + y ˆ j + z ˆ k where the coefficients x , y , and z give the particle’s location along the coor- dinate axes and relative to the origin. As a particle moves, its position vector changes in such a way that the vector always extends to the particle from the origin. If the position vector changes—say, from vector r 1 to vector r 2 during a certain time interval—then the particle’s displacement Δ vector r during that time interval is Δ vector r = vector r 2 − vector r 1 or as Δ vector r = ( x 2 − x 1 )ˆ ı + ( y 2 − y 1 ) ˆ + ( z 2 − z 1 ) ˆ k = Δ x ˆ ı + Δ y ˆ + Δ z ˆ k where coordinates ( x 1 , y 1 , z 1 ) correspond to position vector vector r 1 , coordinates ( x 2 , y 2 , z 2 ) correspond to position vector vector r 2 and Δ x = x 2 − x 1 , Δ y = y 2 − y 1 , Δ z = z 2 − z 1 . 1 4.2 Average Velocity and Instantaneous Velocity Average velocity is vectorv avg = vector r 2 − vector r 1 t 2 − t 1 = Δ vector r Δ t = Δ x Δ t ˆ ı + Δ y Δ t ˆ + Δ z Δ t ˆ k Instantaneous velocity is vectorv = lim Δ t → Δ vector r Δ t = dvector r dt = dx dt ˆ ı + dy dt ˆ + dz dt ˆ k v x = dx dt , v y = dy dt , v z = dz dt x y O 1 2 Δ vector r vector r 2 vector r 1 Path The direction of the instantaneous velocity...
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lecture-ch4 Motion in Two and Three Dimensions - Contents 4...

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